Circuit Analysis Using Complex Variables by & Jonah Taylor Scott Lew

Overview Introduction Complex Circuit Forms The Complex Ohm’s Law

Introduction: AC Circuits Most common type of circuits in everyday life. Driven by time dependent, usually sinusoidal potential. In circuit analysis we are trying to find the current in the circuit and voltage across components. When dealing with ac circuits this leaves us D.Es and messy algebra.

A Simple Example Applying KVL we get . Schematic of a series RC circuit. Applying KVL we get . Assume voltage has the form Vpcos(ωt) and current has the form Plug in and solve for . Easier with complex exponentials

The Better Way Assume current and voltage have the form and respectively. Plug in and take derivatives: Cancel exponentials and rearrange terms:

The Better Way Isolate the current Realize bottom is a complex number and put it in form where Using this Our final answer is

The Complex Ohm’s Law The Even Better Way Observe the previous equation used earlier: This equation has similarities to the famous Ohm’s Law for resistors: The RC circuit discussed earlier can be generalized to include inductors to the mix. These three components can be generalized to all have resistance called Impedance.

Complex Impedance Resistors Capacitors Inductors Unfortunately, resistors don’t have a complex form: Similar to what was seen in the previous equation: The complex form of an inductor: Real & Imaginary Real part of complex impedance: Resistive Impedance or Resistance. Imaginary part: Reactive Impedance or Reactance, denoted by

Complex Ohm’s Law Series Parallel where The Rules for Circuit Analysis Series Parallel

A Series LRC Circuit Example Following the Series Rule: Combining with Complex Ohm’s Law: Performing similar algebra as the other we have: Circuit Diagram where

A Series LRC Circuit Example Using we obtain: Taking only the real part: With this, we have found the current and have successfully analyzed the AC circuit.

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